« ForrigeFortsett »
Page 16, line 2 from bottom, for that the sides about the right
angle, read the three sides of the A
19, for ratio, their, read ratio of their
9, for point, read points
31, for o's, read 0
23, for it, read its.
30, for plain read plane
THE general objects of the present treatise being to facilitate the study of the Elements of Geometry, as well as to promote the exercise of it amongst the students as much as possible; it seems advisable to show how far the reader may presume on these objects being attained. The former is attempted by giving the propositions of the Elements general demonstrations, which must more than any other contribute to give the reader clear and comprehensive views on the subject of Geometry, and to cultivate his reasoning powers with respect to all parts of human knowledge.
On the subject of viva voce demonstrations, much has been said against its general use, particularly in the higher orders of Mathematics and Physics, and indeed too much could not be said if any good were likely to result from it. Academical reform is too hopeless a case, and one of too extensive a nature to be worth discussing at any length.
In plane Geometry, however, where analytic artifice has no place, and where the reader is seldom if ever obliged to lose sight of the basis on which his reasoning rests, viva voce demonstrations are of powerful effect. They accustom him to clearness and solidity of reasoning, and enable him to take a view of the whole of a subject at once, the discussion of which would bewilder any person not accustomed to such.
The second object is aimed at, by presenting the reader with a collection of Geometrical Theorems, Loci, Porisms, and Problems, carefully selected from the best Geometrical writings, and some of them perhaps never before published. Mutual dependence has been considered as well as relative importance in the way of difficulty. The Demonstrations are not given, first from the great size and consequent expense it would entail upon the work; and secondly, because it would otherwise completely fail in the effect of exercising the student in reasoning for himself, and only accustom him to get off by rote the reasoning of others.
The best system of Geometry that has lately been presented to the public is that of Professor Leslie, as far as he confines himself to the strict limits of plane Geometry. As for the crude, inadequate and false demonstrations that are huddled together at random in the books of Creswell and Bland, the reader is cautioned to avoid them; unless where he may be occasionally recommended to use them as being the works of some abler hands; for it may be said of these writers, as it has been said by an able critic of one of their cotemporaries, that “ their errors only are original."
Perhaps the collection here presented is the most complete and most numerous that has yet appeared; and is by no means to be considered as a mere geometrical exercise : it is a collection of truths, and as such a powerful and necessary portion of human knowledge. They have a still greater use, “ humana' notitia et humana potentia in idem coincident,” says Lord Bacon ; they put
the reader in possession of new resources, new instruments for the discovery of truth, and thus afford means of extending the portion of human knowledge that they are connected with. It is by no means to be understood that Geometry ought to be introduced into the higher branches of Mathematics; this would be a use of it totally out of its proper line, a use of it that could only be attempted in modern times by a Scotch Professor, anxious to lay hold on a claim to originality, one way or other, in spite of his stars. Let us recollect that the Theorems of Geometry are translatable into analytic language ; that when so 'translated they are capable of all the enlargement and extension that analysis in general admits, and that in this way they may be said to exercise an indirect influence over the whole of Mathematics, of unbounded power and extensive use. marks are not made at random; they must be acknowledged true by all acquainted with mathematical pursuits. How many Theorems, originally geometrical, have been generalised and carried to the highest perfection, and applied to the most important uses by analytic writers, that perhaps would otherwise have never been thought of? Need we instance in the celebrated Theorems of Vieta that have led to the summation of series of sines and cosines of multiple arcs, the formation by induction of the expressions for the tangents of multiple arcs by. De Lagny, and Cotes's celebrated Theorem, which we may be certain was ultimately founded on geometrical reasoning, and is announced by the Author as a Geometrical Theorem.
On such theorems as these the whole of Analytic science is founded, and must continue to be founded until analytic science is able to manage elementary truths, until it is able to bring under its dominion the principles of Trigonometry, if ever this æra shall arrive.* Every attempt then to decry Geometry, must be an attempt to put down all Mathematics, no less than that which would attempt to reduce all to plane Geometry. Every person that would make either attempt is no Mathematician, is a pretender, one that has fastened to some particular branch of Mathematics, rings endless changes upon the principles of it, and calls that Science. Such persons, however, can do but little mischief, as they in general have but small influence with others, and the best use that can be made of them, is to set them to worry one another. He only is the true Mathematician who is ready to give each-its proper share of weight, to leave them their full influence as an united and combined instrument of calculation, and not to attempt to sever that union which affords them their chief strength.
Much more might be said on this subject, but it would be inconsistent with the plan of this little tract; so we shall forbear for the present, requesting the reader's indulgence for any oversights he may meet with.
# In a book entitled, Memoirs of the Analytic Society, published at Cambridge in 1813, after proving the 47th of Euclid's first book by the theory of functions, the following remark occurs ; " It is only by this way of proceeding, or some analogous one, that we can ever hope to see the elementary principles of Trigonometry brought under the dominion of Analysis. It may suffice to have thrown out a hint that may be followed up at some future opportunity.”